I^n中的孔
Holes in I^n
摘要：
设F是特征不为2的任意域。我们把F的Witt环记为W（F），它由F上所有各向异性二次型的同构类组成。对于W（F）的任何元素x，维数dim x定义为表示x的二次型的维数。所有偶数维的元素形成一个理想，记为I（F）。环W（F）通过这个理想的幂I（F）^n的过滤在二次型的代数理论中起着基本的作用。最近由Voevodsky和Orlov-Vishik-Voevodsky证明的Milnor群描述了这个滤子的连续同调群I（F）^n/I（F）^{n+1}，将它们与域F的伽罗瓦上同调群和模2的Milnor K-群等同起来。在这篇文章中，我们给出了一个关于I（F）^n的一个不同的老问题的完整答案，这个问题是关于在I（F）^n中x的dim x的可能值。更精确地说，对任意正整数n，我们证明了对于I（F）^n和所有F中的所有x，所有dim x的集合dim I^n由2^{n+1}-2 ^i组成，i= 1，2，.，n+1以及所有大于或等于2^{n+1}的偶数。以前关于dim I^n的部分信息包括经典的阿拉森-菲斯特定理，该定理认为dim I^n中没有0到2^n之间的整数，以及最近的维希克定理，该定理对2^n到2^n+2^{n-1}之间的整数也有相同的说法（n=3的情况是菲斯特的，n=4的情况是霍夫曼的）。我们的证明是基于在周组的权力的射影二次（涉及Steenrod操作）的计算，开发的方法也可以应用于其他类型的代数簇。
英文摘要：
Let F be an arbitrary field of characteristic not 2. We write W(F) for the Witt ring of F, consisting of the isomorphism classes of all anisotropic quadratic forms over F. For any element x of W(F), dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the powers I(F)^n of this ideal plays a fundamental role in the algebraic theory of quadratic forms. The Milnor conjectures, recently proved by Voevodsky and Orlov-Vishik-Voevodsky, describe the successive quotients I(F)^n/I(F)^{n+1} of this filtration, identifying them with Galois cohomology groups and with the Milnor K-groups modulo 2 of the field F. In the present article we give a complete answer to a different old-standing question concerning I(F)^n, asking about the possible values of dim x for x in I(F)^n. More precisely, for any positive integer n, we prove that the set dim I^n of all dim x for all x in I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all even integers greater or equal to 2^{n+1}. Previously available partial informations on dim I^n include the classical Arason-Pfister theorem, saying that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's theorem, saying the same on the integers between 2^n and 2^n+2^{n-1} (the case n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in Chow groups of powers of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.
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